For a character living in two dimensions, grasping the idea of life in 3D can be tough - especially if the character in question is Homer Simpson. But one Halloween episode of The Simpsons forces him to confront the concept, and gives viewers a mathematical workout too.

Eighteen years ago, the writers of The Simpsons celebrated Halloween with one of their traditional Treehouse of Horror episodes, which included three short stories. After Attack of the 50-Foot Eyesores and Nightmare on Evergreen Terrace, viewers were treated to Homer³, which contains the most intense five minutes of mathematics ever to appear on prime time television.

For many people, mathematics is more terrifying than being attacked by an army of zombies, werewolves and vampires, but in this case the various equations were included because the writer of Homer³, David S Cohen, is a fan of numbers.

The storyline involves Homer diving through a portal and entering a peculiar three-dimensional universe. When he senses his new extra-dimensionality, Homer says: "What's going on here? I'm so bulgy. My stomach sticks way out in front."

It is a particularly memorable plot, because the animation changes from the classic Springfield-style to futuristic computer graphics when Homer enters the portal. It is within this computer-generated landscape that eagle-eyed number nerds (like myself) can discover Cohen's mathematical nuggets.

For example, in one scene, the letters P and NP can be seen over Homer's right shoulder. Although these three letters would have made no sense to most viewers, they are a deliberate nod towards a statement about one of the most important unsolved problems in theoretical computer science. Indeed, this is such a weighty puzzle that there is a reward of $1m (£623,000) for whoever solves the mystery.

P stands for polynomial and NP for non-deterministic polynomial. One way to think about these terms is that P-type problems are essentially easy to solve, while NP-type problems are difficult.

More maths in Homer³

The 3D landscape includes a rearrangement of an equation known as Euler's identity. This is arguably the most beautiful equation in history, because it includes five fundamental ideas within mathematics, namely e, i, π, 1 and 0

An apparently random series of hexadecimal digits (base 16) is actually a message written in ASCII, a notation for turning numbers into letters. It reads "Frink rules!"

A Utah teapot makes a cameo appearance in Homer³ (and in Toy Story and Monsters, Inc). This is a standard object used to test and compare different ways of mathematically modelling 3D objects

The question for mathematicians is whether NP-type problems are fundamentally hard, or whether there might be a trick that could turn these hard problems into easy P-type problems.

It is surprising to find a reference to P- and NP-type problems in a television sitcom, but not when the writer is Cohen, because he studied them while doing his master's degree at the University of California in Berkeley.

Meanwhile, back in Springfield, Professor Frink is giving Chief Wiggum and others an impromptu introduction to the mathematics of higher dimensions in order to explain what happened to Homer when he disappeared through the portal. Of course, the notion of a third dimension is baffling from Chief Wiggum's two-dimensional perspective, so Frink draws a diagram:

Professor Frink: Here is an ordinary square.

Chief Wiggum: Whoa, whoa! Slow down, egghead!

Professor Frink: But suppose we extend the square beyond the two dimensions of our universe along the hypothetical z-axis... There.

Everyone: [gasps]

Professor Frink: This forms a three-dimensional object known as a cube, or a Frinkahedron in honour of its discoverer.

Perhaps the most striking piece of mathematics in Homer³ is a reference to Fermat's last theorem, the most notorious problem in the history of mathematics.

According to the 17th Century French mathematician Pierre de Fermat, it is impossible to find two numbers raised to the 12th power that equal a third number raised to the 12th power, but this equation from Homer³ seems to defy Fermat's claim:

1,782¹² + 1,841¹² = 1,922¹²

Indeed, a quick check on your phone calculator will suggest that Homer has proven Fermat to be wrong. However, this is not an actual solution, but merely a so-called near-miss solution, which means that only a highly accurate calculating device (with more decimal places than the display on the average calculator) will see the slight imbalance in the equation. In other words, Cohen was playing a Halloween prank on viewers.

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The year 1913 marked the beginning of an extraordinary relationship between an impoverished Indian clerk and a Cambridge don. A century later, their remarkable friendship has left its mark in the strangest of places, namely in the animated series Futurama.

It was this bit of mathematical tomfoolery that first caught my attention when I watched Homer³, and I only spotted the other references on second and third viewings. In time, I realised that there were several mathematical writers on The Simpsons, and indeed there are many other episodes alluding to number theory and geometry.

However, as yet I have failed to discover any appearance by the number 1,000,000,000,000,066,600,000,000,000,001. This is a great shame, as it is a terrifying number and would be particularly suitable for a Halloween episode of The Simpsons.

It is known as Belphegor's prime. It is a prime number because nothing will divide into it except one and the number itself, and it is named after Belphegor (one of the seven princes of hell) because it has 666 at its heart and 13 zeroes either side of the number of the beast.

Perhaps the writers of The Simpsons are saving it for next year's Treehouse of Horror episode.

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